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Optimal Error Bounds in Normal and Edgeworth Approximation of Symmetric Binomial and Related Laws

  • This thesis explores local and global normal and Edgeworth approximations for symmetric binomial distributions. Further, it examines the normal approximation of convolution powers of continuous and discrete uniform distributions. We obtain the optimal constant in the local central limit theorem for symmetric binomial distributions and its analogs in higher-order Edgeworth approximation. Further, we offer a novel proof for the known optimal constant in the global central limit theorem for symmetric binomial distributions using Fourier inversion. We also consider the effect of simple continuity correction in the global central limit theorem for symmetric binomial distributions. Here, and in higher-order Edgeworth approximation, we found optimal constants and asymptotically sharp bounds on the approximation error. Furthermore, we prove asymptotically sharp bounds on the error in the local case of a relative normal approximation to symmetric binomial distributions. Additionally, we provide asymptotically sharp bounds on the approximation error in the local central limit theorem for convolution powers of continuous and discrete uniform distributions. Our methods include Fourier inversion formulae, explicit inequalities, and Edgeworth expansions, some of which may be of independent interest.

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Author:Patrick van Nerven
URN:urn:nbn:de:hbz:385-1-23767
Referee:Lutz Mattner, Bero Roos, Thorsten Neuschel
Advisor:Lutz Mattner
Document Type:Doctoral Thesis
Language:English
Date of completion:2024/11/03
Publishing institution:Universität Trier
Granting institution:Universität Trier, Fachbereich 4
Date of final exam:2024/04/25
Release Date:2024/11/14
Tag:approximation binomial normal edgeworth local global higher order
Number of pages:v, 100 Seiten
First page:i
Last page:100
Institutes:Fachbereich 4
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoCC BY-ND: Creative-Commons-Lizenz 4.0 International

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